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**Modelling Aquatic Rates In Natural Ecosystems BIOL471**

Prey dependent responses School of Biological Sciences © 2001 School of Biological Sciences, University of Liverpool

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**Prey-dependent responses**

Solomon (1949) separated consumer response to prey density into 2 types: Functional the consumption rate of individual consumers with respect to resource density Numerical the per capita reproductive rate with resource density Holling (1959) identified 3 types of functional responses Ingestion Prey Growth rate Prey + -

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**Prey-dependent responses**

Type I (linear) response The attack rate of the individual consumer increases linearly with prey density but then reaches a constant value when the consumer is satiated Ingestion Diatoms (ml-1) Prey

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**Prey-dependent responses**

Type II (cyrtoid) functional response The attack rate increases at a decreasing rate with prey density until it becomes constant at satiation Cyrtoid responses are typical of predators that specialise on one or a few prey Ingestion Prey

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**Prey-dependent responses**

Type III (sigmoid) functional response The attack rate accelerates at first and then decelerates towards satiation Sigmoid responses are typical of generalists that switch from one prey species to another and/or increase their feeding when resources are abundant Ingestion Prey

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**C = aH 1+ThaH Prey-dependent responses The “Disk” Equation**

Holling (1959) derived a mechanistic mathematical model for the Type II response from experiments in which blindfolded people acted as predators by searching a table top with their finger tips for sandpaper disk prey We can derive the disk equation C = 1+ThaH aH

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**Ttotal = Tsearch + Thandle**

Prey-dependent responses Let us assume that there are two activities involved in consuming a prey: Searching for the prey Handling or processing the prey Then the total time (Ttotal) to capture prey is: Ttotal = Tsearch + Thandle But we want to know the prey consumed (Hc) over time Ttotal Then we can determine a consumption rate C (HT-1)

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**Thandle = Hc*Th Prey-dependent responses**

A predator will consume Hc prey in time T There is a handling time (Th) Total time handling (Thandle) will be the product of handling time (Th) and the prey consumed (Hc) Thandle = Hc*Th Where: Hc prey consumed (H) Th is the handling time of one prey (TH-1)

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**Hc = a * H * Tsearch Hc Tsearch = aH Prey-dependent responses**

Also a predator will capture a number of prey (Hc) over a searching time (Tsearch) if it has a constant searching rate (a) But this will depend on how many prey (H) there are This can be expressed as Hc = a * H * Tsearch Where a is the searching rate (L2T-1) H is the prey density (HL-2) and Tsearch = Hc aH

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**Ttotal = Tsearch + Thandle**

Prey-dependent responses We can now substitute into Ttotal = Tsearch + Thandle Thandle = HcTh Tsearch = Hc aH Ttotal = + HcTh Hc aH

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**= = Prey-dependent responses**

We can now rearrange this equation to solve for Hc T = + HcTh Hc aH Hc = prey captured (H) a is the searching rate (L2T-1) H is the prey density (HL-2) Th is the handling time (TH-1) T is total time (T) = 1 C is consumption rate (HT-1) T = + HcThaH aH Hc T = Hc + HcThaH aH = 1+ThaH aH Hc T T = Hc(1+ThaH) aH = 1+ThaH aH C Hc = 1+ThaH aHT

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**Prey-dependent responses**

We can now rearrange this equation to solve for Ha C= 1+ThaH aH C = ( H) H 1 Th aTh C = (1+ThaH) aH 1 a If = Cmax 1 Th aTh If = k C = ( +ThH) H 1 a C= ( +ThH) H 1 a Th C = k+ H CmaxH

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**Prey-dependent responses**

Cmax C = k+ H CmaxH C k Cmax 1 2 Cmax is the maximum grazing rate k is the half-saturation constant H

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**The experimental approach**

We run experiments to determine grazing rates Then we can use these rates in models We can also use these experiments to determine constants that tell us about the biology of the organisms The formula we just examined was based on biological mechanisms It is called a mechanistic equation Other models that are not based on mechanisms are called phenomenological equations C H C = k+ H CmaxH

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**a is the searching rate (L3T-1) Th is the handling time (TH-1)**

Prey-dependent responses + a Th a is the searching rate (L3T-1) Th is the handling time (TH-1)

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**Prey-dependent responses**

H = Ct Beads (H) C= H t Time (t)

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**Prey-dependent responses**

H = Ct Beads (H) C= H t Time (t)

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**Prey-dependent responses**

C = k+ H CmaxH C (HT-1) [H ] (HL-3)

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**The experimental approach**

Determine the rate of eating Smarties Provide various amounts to students Experiment (consume Smarties) Plot data Determine a mechanistic equation Examine biological parameters

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**Prey-dependent responses**

C = k+ H CmaxH Smarties min-1 Smarties density (H L-2)

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*** The experimental approach a the searching rate (L2T-1)**

The time it takes to touch your nose and then the desk the area encounter Th the handling time (TH-1) This is the time it takes you to eat a Smarties *

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